The behavior on fundamental group of a free pro-homotopy equivalence. II
In an earlier paper [ibid. 13, 239-253 (1982; Zbl 0481.55007)], the authors studied the question whether a morphism in pro- (pointed homotopy), which is an isomorphism in pro- (homotopy), is also an isomorphism in pro- (pointed homotopy). An affirmative answer to this question was given under suitable hypotheses. The present note continues this study and it is proved that the answer to this question is again affirmative provided the \(pro\)-\(\pi\) \({}_ 1\) of the target has pro- finite index. More specifically, the following is shown: If f: (X,x)\(\to (Y,y)\) is a morphism of pointed shape of pointed compacta such that \(f: X\to Y\) is an unpointed shape isomorphism, then f is a pointed shape isomorphism provided \(pro\)-\(\pi\) \({}_ 1(Y,y)\) has pro-finite index. The notion of pro-finite index is discussed.
scientific article; zbMATH DE number 3955982